Advanced Rolling Theory

Rolling theory and its applications comprise a broad spectrum of disciplines and interests. The discussion of this complex and varied subject will be partitioned into three seminars.

Part 1 – Mechanics of the Material and Roll Bite

This seminar provides a detailed understanding of how the material behaves and deforms (in elastic and plastic modes) during its stress-loaded transit through the roll bite.

• At the atomic level, slip planes, dislocation migration, grain boundary interactions and Cottrell end-pinning actions (Lüders band formation) are considered.

• At a macro scale, the biaxial yield criteria, plastic flow dynamics and work hardening mechanisms are derived and examined to explore the underlying physics and dynamics of plastic deformation.
  Included are the full analytic derivations and explanations of:

  • Cauchy stress tensors.

  • Biaxial yield criteria and deviatoric invariants.

  • Tresca and Von Mises yield criteria.

  • Plastic/elastic deformation conditions (in three dimensions) at the exit plane.

• The mechanics of the mill/material interaction are discussed to establish a preliminary understanding of the event sequence, phases of deformation and the forces experienced in the roll bite.

The seminar concludes by introducing the basic components of the roll bite’s longitudinal pressure distribution.

Part 2 – Analytic Models of the Cold Rolling Process

Using first-principles physics and the previously derived material characterizations, this seminar dissects the longitudinal roll bite to establish what’s going on and develop detailed mathematical models of the circular and non-circular arc representations of the conditions.

• Prerequisite definitions and analytic derivations of the roll bite’s classical terms, parameters and conditions are presented, including: thickness progression, arbitrary points/planes within the roll bite, bite angles and arc length of contact.

• The material’s flow dynamics and strain rates are discussed, and the amplitude/polarity of the contact shearing/friction forces are considered.

• Rolling speed variation of the friction and its slip/stick behavior are characterized and discussed.

• Methods of describing the nature of elastic roll flattening are presented, along with their validity during certain classes of rolling is discussed.

• A more detailed understanding of the longitudinal pressure distribution is assembled, and an examination of its reactions/sensitivities to rolling conditions and parameter variations is conducted.

Combining these findings leads to the full analytic derivations and explanations of historically significant relationships:

• Von Karman’s horizontal equilibrium equation.

• Orowan’s equation.

Using these foundational relationships, we can assemble detailed analytic models capable of predicting the conditions and events of the rolling process.

• Circular Arc – An examination of cases involving moderate reductions with slip-type friction in a circular arc, roll flattened geometry. This discussion will include full analytic derivations and explanations of historically significant relationships:

  • ♦ Hitchcock Roll Flattening.	♦ Roberts.
    ♦ Bland and Ford.	♦ Stone/Nadia.
    ♦ Bryant and Osborn.

  • Non-Circular Arc – An examination of the conditions that cause the classical circular arc assumption to no longer be valid, requiring an alternate approach.

    • Primitive indentation conditions from non-elliptic pressure distributions are presented.

    • The need for a more dexterous, accurate characterization of roll flattening is provided by a Hertzian formulation.

    • Returning to Orowan’s equation and considering slip/stick friction conditions in the presence of work-hardened yield stress (in shear).

    • Several contemporary non-circular arc models are presented:

      ♦ Hertzian Roll Flattening.	♦ Grimble.
      ♦ Jortner.	♦ Le and Sutcliffe.
      ♦ Alexander.	♦ Fleck and Johnson.

• This discussion concludes by examining how advanced models are applicable to specific rolling conditions (such as temper/skinpass rolling), along with mill/process design, selection and sizing.

Part 3 – Practical Concerns, Model Adaption and Applications

This closing seminar focuses on the use of these models, where theory meets reality in the form of model prediction inaccuracies, and how model adaption can be employed to improve performance and stability.

• Conceptual illustrations are used to explain the nature of mathematical modeling efforts, their objectives, specifications and predictive power.

• Trade-offs between model accuracy and model complexity are discussed and correlated with concerns about model stability and convergence failures.

• The concept of model adaption is introduced and short-term/long-term influences are shown to improve predictive accuracy, while also permitting model simplification to address stability/convergence concerns.

• Applications of these models are considered with an initial focus on multi-run strategies for setup/threading and mill capabilities envelope evaluation.

• Transverse deflection modeling/indications are briefly discussed to provide insight into how coupled modeling strategies optimize pass scheduling and setup conditions.

This seminar concludes by looking at other places where theory and modeling can assist in: process design, off-line simulation studies, what-if scenarios, performance and production prediction, multi-stage reduction/annealing planning, mill/equipment sizing and selection, operational assistance, and guidance in problem resolution.

 Automatic Gauge Control (AGC)

The accuracy, precision and consistency (over the product mix) of the rolled strip’s thickness are critical quality indicators, requiring high-performance, real-time control of the mill (i.e., automatic gauge control). The discussion of this important subject will be partitioned into two seminars.

Part 1 – The Theory, Mechanics and Sensitivities of the Gauge Control Problem

This seminar steps beyond the simple block diagrams of introductory AGC discussions and looks deeply into what’s really going on within this control problem, how all this works, and the architectures of the systems that implement the on-line solutions in real time.

• Examination of the overall control problem, the sources of gauge deviations, the objectives and the involved components (i.e., the mill, roll stack, sensors, actuators and controller(s)).

• Three baseline models/components are derived:

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  • ♦  Generalized Actuator Model.

    ♦ Mill Modulus Component.

    ♦ 4-Quadrant Roll Stack/Force Transmission Model.

• Identify and analyze what happens when an actuator is adjusted, and how that action may disturb other control systems in the mill (e.g., tension controls). This introduces a pair of additional models:

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  • ♦ Mass Flow Dynamics Model.

    ♦ Mass Flow Disturbance Model.

• Introduction of the sequential “dance steps” taken while executing the gauge control solution in real time:

  • Identifying gauge deviations.

  • Determining corrective actions.

  • Applying limits associated with the rolling conditions.

  • Properly timing/applying corrective adjustments to the mill/material.

• Philosophical discussion on length-based, time-based, speed-based and hybrid systems.

• Dissection of the controller and each of its subsystems:

  • Sensor signal conditioning, signal estimation, self-calibration.

  • Strip tracking, transport buffering and critical signal extraction.

  • Internal model of the mill/material interactions and actuator dynamics.

  • On-line/in-pass material adaption.

  • Correction determination and limiting.

  • Actuation signal shaping and timing.

  • Non-interactive compensation with adjacent equipment.

• Introduction of the “internal model” concept and its central role in determining the needed corrective adjustments and non-interactive compensation. A case study of a typical model involving a cascade is paired with:

  • A semi-sophisticated off-line model based on the physics/mechanics of the mill/material interactions, for a fully instrumented, single-stand arrangement.

  • An on-line, small-signal, perturbation model whose sensitivities are derived from a linearization of the off-line model, localized to the current operating point.

• Discussion of the critical timing and scaling of actuator adjustments to:

  • Cause the imparted corrective adjustments to the roll bite’s plastic deformation process.

  • Coincide with the arrival of the deviated material in the roll bite.

  • Reduce or eliminate gauge deviations in the roll/exit material.

This seminar closes by:

• Determining the sensitivity functions of the primary and secondary reactions, which will directly impact the design of the control strategies.

• Determining the gain structures as functions of the material geometry, resistance to deformation, mill setup and rolling conditions.

Part 2 – AGC Mode Details, Special Functions and Tandem Mill Controls

This seminar is composed of two distinct parts, with the first extending the previous findings to the design/assembly of AGC systems:

• Detailed examination of the classical AGC modes.

• Special functions and adaptions that provide the ability to accommodate the entire product mix and range of rolling conditions.

• Practical considerations and methods of cooperation with other mill systems are also discussed, with an emphasis on achieving improved performance via cooperative pass scheduling and non-interaction with the shape controls and strip tension systems.

• Temper rolling, skinpassing and surface conditioning control modes are included.

This seminar’s second part focuses on gauge and tension control in various tandem mill arrangements.

• Typical tandem mill arrangements and operations.

• Classical sensing and actuation arrangements.

• Reduced/constrained sensing configurations (typically experienced with failed/off-line instrumentation or reduced-cost scenarios).

• Pivot stand strategies are discussed and configurations involving last-stand surface conditioning (EDT) are considered.

• Open-gap and closed-gap threading/tail-out strategies are presented and discussed.

Time permitting, this seminar may conclude with a detailed discussion on how to evaluate on-line/off-line gauge control performance.

Shape/Flatness Measurement and Control

The concepts and realities of shape, flatness and profile are closely related, but are fundamentally different in the way they describe and characterize the same underlying phenomena: The material’s reaction to the presence of plastically induced, non-uniform strain, embedded within the solid body of the strip.
Unfortunately, in much of the common language and discussions, the terms shape, flatness and profile are used arbitrarily and interchangeably, leading to confusion and misunderstandings. Further, the flatness/shape of the rolled strip (and its consistency) are critical performance, specification and quality indicators that require a combination of mill scheduling and setup, along with shape measurement/control.
There’s a lot involved in this process/control problem, and the discussion of this multi-faceted subject will be partitioned into two seminars, starting with an analytic examination of the underlying phenomena, then progressing to detailed examination of how one measures and controls shape to render the desired flatness.

Part 1 – Understanding and Characterizing Flatness, Shape and Profile

This seminar will specifically define these terms and their interrelationships, along with an examination of the events and conditions that form residual stress and flatness defects, and how they can be characterized, compared and specified.

• Flatness — The concept of “flatness” is introduced with a focus on understanding and analytically characterizing the geometry of manifest buckling.  

  • An extensive tour is then conducted of classical, manifest buckling defects forming in the longitudinal, transverse and through-thickness directions.

  • The physics/mechanics of “buckling” and the buckling threshold are examined showing the fundamental relationship with compressive strain, and the consequences of tensioned and non-tension conditions.

  • Methods of describing and characterizing flatness defects in terms of their manifest geometries.

• Profile — The interaction between the transverse profiles of the strip and roll gap is examined.

  • Definition of the term “crown” and delineated for the rolls versus the strip.

  • The concept of crown ratio is introduced and this is also carefully delineated for the strip versus the roll gap.

  • Through illustrated examples, the root-cause mechanism of relatively simple shape/flatness defect formation is examined, and the influence of tensioned and non-tension conditions are considered.

• Shape — The concept of “shape” is introduced.

  • The solid body whole, relaxed strand depiction and equal length plane are used to define and characterize the “shape” via plastic-induced differential strand length, strain, and stress.

  • The relation of non-uniform plastic deformation and shape are discussed, and the specific definitions of residual strain and stress are introduced.

  • The influence of uniform and non-uniform tension are examined, along with how they can mislead shape indications.

• The multi-step sequence of events and conditions that cause the formation of flatness defects is presented.

• The tour of flatness defects is revisited, and for each defect, the exit plane conditions are reconstructed in terms of relaxed strand depictions (shape).

  • Using these results and a knowledge of the transverse uniformity of the applied exit tension, the nature and amplitude of the strip/roll bite profile mismatch (and/or possible mechanical misalignment, planar and through-thickness shearing, thermal gradients, etc.) are identified as the root-cause contributors of the experienced defects.

• The fundamental interrelations of flatness, profile and shape are analytically derived to show where the many popular/published depictions of these terms originate.

• The term “I-Unit” is defined in fundamental strain, then extended to stress, flatness and profile. As a bonus, the rarely seen first-principles derivations and clarifications of several key equations/relations will be included:

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  • ♦ Detailed derivation of the “popular” I-Unit equation for flatness.

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  • A comparative study is conducted to show the attributes of each depiction and how they individually (and correctly) characterize the exact same physical phenomena, but in different and misleading ways.

• Methods of characterizing the potentially complex natures of the flatness, profile and shape transverse waveform patterns are discussed.

  • Arbitrary waveform patterns are reduced to their simplified spatial curvature components by parameter decomposition within an orthogonal polynomial basis.

  • Common framework for depicting shape/flatness conditions.

  • The “circle” of shape/flatness defects.

This seminar closes with a series of case studies that consider other common flatness defects, which form in the transverse and through-thickness directions (e.g., coil-set and crossbow).

• Using the previous developments, findings and techniques, these off-axis defects are shown to behave in a manner identical to the longitudinal defects. However, their manifest buckling forms in orthogonal directions, from causes outside the roll bite.

Part 2 – Shape/Flatness Measurement, Control and Targeting

This multi-faceted seminar discusses and develops methods of measuring and controlling the strip’s on-line shape, and this is used to achieve the desired off-line (delivered) strip flatness conditions (which may not be “still-water” flat).

• Introduction and examination of the predominant issue of off-line flatness control, and a preliminary glimpse of the overall measurement/control architecture.

• Examination of the underlying concept of extracting the strip’s residual stress pattern from measurement of the strip’s transverse stress profile while under tension.

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  • ♦ Derivation of the governing analytic expressions that transform physically measurable conditions to accurate depictions of shape.

•   Detailed “tour” of contemporary shape/flatness measurement techniques, sensing technologies, installed locations and commercially available systems.

  • Review of the fundamental limitations of the systems and uncertainties of the raw measurements (e.g., partially covered edge zones, sensor accuracy, overall system accuracy, etc.)

• Shape measurement errors stemming from mechanical misalignments, non-uniform applied tension, misregistration of the strip on the sensing array, wrap angle uncertainty, thermal gradients across the strip, etc.) are discussed and methods of compensation/correction are presented.

• Additional signal conditioning, smoothing filtration and masking techniques are introduced to produce more accurate measurements and improve the overall system stability.

• Step-by-step procedure that accepts raw sensor signals and rendering the final shape/flatness measurements.

• Review of the nature and characterization of transverse patterns/waveform structures.

  • Parameter decomposition of the measured shape waveform, to render a simplified depiction in terms of its underlying curvature components (for use in on-line characterization and advanced control strategies).

• Presentation of various methods of displaying and visualizing the shape/flatness measurements, along with how to avoid the indications misrepresenting the actual conditions, and how to misinterpret what the displays are saying.

  • Special emphasis is given to interpreting the vertical polarity of the displayed indications and what they depict in terms of the actual conditions.

The discussion shifts to the architecture and analytic strategies employed in on-line shape control systems, with a special emphasis on vertical stack mill arrangements. Shape control strategies for 20-high/cluster mills will be covered in the Special Topic seminars.

• Review of the general architecture and components of a typical on-line shape control system.

• Review of the shape adjustment influence characteristics and temporal dynamics of the mill’s individual shape actuators (i.e., tilt, roll bending, zonal sprays).

  • Special interest as to each shape actuator’s adjustment waveform.

• The natural force-loaded, transverse deflection of the roll stack is examined and analytically described. This includes the mechanical crowning (or complex curvatures) ground into the mill’s rolls to offer some degree of deflection compensation.

• Combining these characterizations, a composite model of the mill and shape actuation is formed, which represents deviation responses in the roll bite’s profile or in the measured shape. This model serves as the “internal model” embedded within the controller module of the closed-loop system.

• The overall closed-loop controller is assembled, with key observations and discussions on the signal flow and coupled, multi-actuator correction distribution.

• The shape target is defined, with included discussions on its selections, pass schedule coordination, and how strip location can be used to provide proper registration with the mill’s actuation.

• The shape error is defined, and strategies for decimating and distributing its contents are discussed.

• Detailed examination of the completed shape control system, including higher-level sensitivity adjustments, on-line parameter identification and the necessary adaptions to accommodate the entire product mix.

• Introduction and analysis of methods for measuring and indicating the on-line shape control performance, and the differences in the specification of on-line shape versus off-line flatness.

This seminar concludes by returning to the off-line flatness control problem and discussing how to employ various shape targeting and pass scheduling strategies to achieve the desired strip flatness characteristics, in the presence of post-rolling/post-coiling conditions that may alter the on-line results.